Electromagnetism

Faraday's Law of Induction

A changing magnetic flux through a circuit induces an EMF — the foundation of generators and transformers

Faraday's law of induction: the EMF (electromotive force) induced in any closed loop equals the negative rate of change of magnetic flux through any surface bounded by the loop: EMF = −dΦ_B/dt, where Φ_B = ∫ B·dA. Discovered by Michael Faraday in 1831 (independently by Joseph Henry slightly earlier, but unpublished). Three sources of dΦ_B/dt: (1) changing B (with stationary loop), (2) moving loop in static B (motional EMF), (3) deforming loop. Lenz's law (the minus sign): the induced current opposes the change. Differential form (Maxwell): ∇ × E = −∂B/∂t. Powers all electric generators (90% of world electricity), transformers (steps voltage up/down), induction stoves, MRI scanners, wireless charging (Qi, 5W-15W typical). Faraday's drawings of magnetic flux were also the conceptual seed for modern field theory.

  • Integral formEMF = −dΦ_B/dt
  • FluxΦ_B = ∫ B·dA
  • AuthorsFaraday 1831 (Henry independent)
  • Differential form∇×E = −∂B/∂t
  • Lenz's lawInduced current opposes change
  • Civilizational impact~90% of world electricity from generators

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Why Faraday matters

  • Generators. Every coal, gas, hydro, nuclear, and wind power plant on Earth ultimately drives a magnet (or coil) past windings — Faraday's law converts the mechanical rotation of the prime mover into electrical energy. This single physical mechanism produces about 90% of all electricity globally.
  • Transformers. The shared core flux between primary and secondary windings, governed by Faraday's law, is what enables the AC grid: step-up to 400kV for transmission to cut I²R losses, step-down to 230V for distribution, and step-down further inside power supplies.
  • Induction motors. Three-phase AC creates a rotating magnetic field; the rotor's conducting bars see changing flux and develop induced currents that interact with the stator field to produce torque. Tesla's 1888 induction motor — Faraday's law in mechanical reverse.
  • Wireless charging. Qi pads (typically 5–15 W) and emerging EV pads (up to 11 kW) couple two coils across an air gap; the changing primary flux induces secondary EMF — same principle as a transformer, just with a less efficient air-core path.
  • MRI gradient pulses. Rapidly switching the gradient coils on and off creates large dB/dt that induces voltages in everything nearby — patient nerves (peripheral nerve stimulation thresholds), implant leads, and shielding. Designing for safe dB/dt is a Faraday-law engineering problem.
  • Induction stoves. A coil under the cooktop carries 20–50 kHz AC; the changing flux induces eddy currents directly in a ferromagnetic pan, which dissipate as heat. The cooktop surface stays cool because it is non-magnetic glass-ceramic.
  • RFID and contactless cards. The reader's coil radiates a 13.56 MHz magnetic near-field; the card's printed-coil antenna picks up induced EMF, rectifies it for power, and modulates a return signal. No battery needed in the card.

Common misconceptions

  • "Lenz is a separate law." Lenz's law is just the minus sign in Faraday's law expressed in plain language. Together they encode energy conservation; you can't have one without the other.
  • "Current must flow." Faraday's law produces an EMF — a voltage — even if no current flows. An open-circuited coil with changing flux through it has measurable terminal voltage but zero current. EMF is the driving force; current is the response (set by impedance).
  • "Instantaneous response." The EMF responds instantaneously to dΦ/dt, but the current does not — circuit inductance and resistance set a time constant L/R during which current builds up. In a heavily inductive circuit, current may lag the flux change by milliseconds.
  • "Only changing B counts." Three different mechanisms can change flux: changing B with a fixed loop, moving the loop through a static B (motional EMF), or deforming the loop's shape. All three produce identical EMF for identical dΦ/dt.
  • "Faraday's law gives only emf magnitude." The signed version, with proper choice of loop orientation and surface normal, also gives the direction of the induced current via Lenz's law — no need for a separate hand-rule lookup once you've fixed the sign convention.
  • "Mutual induction is different physics." Mutual inductance M between two circuits is just Faraday applied across them: ε₂ = −M dI₁/dt. Same law, different geometry. Self-inductance L is the special case where the inducing and induced loops are the same.

Canonical Faraday calculations

  • Sliding rod on rails. Length L, velocity v, perpendicular B: ε = BLv.
  • Rotating rectangular loop in uniform B. Area A, angular frequency ω: Φ = BA cos(ωt), so ε = BAω sin(ωt) — the basic AC generator output.
  • N-turn coil. EMF multiplies by the number of turns: ε = −N dΦ/dt. This is why generator armatures and transformer windings stack hundreds of turns.
  • Self-inductance L. For any coil, Φ = LI when self-flux dominates, so ε = −L dI/dt. Inductors store energy (½LI²) by virtue of this back-EMF.
  • Mutual inductance M. Two coupled coils: ε₂ = −M dI₁/dt. Coupling coefficient k = M/√(L₁L₂) ranges from 0 (no shared flux) to 1 (every line links both).
  • Maxwell's curl form. ∇ × E = −∂B/∂t. The differential statement: a time-varying B is the source of a curling E field, even in vacuum where no charges or currents are present.

Frequently asked questions

What is magnetic flux Φ_B?

Magnetic flux Φ_B is the surface integral of B·dA over any open surface bounded by a closed loop. Geometrically, it counts the number of B field lines piercing the loop. For a uniform B perpendicular to a flat loop of area A, Φ_B = BA; tilt the loop by angle θ from perpendicular, and Φ_B = BA cosθ. The SI unit is the weber (Wb = T·m²); 1 Wb piercing a 1-turn loop in 1 second induces 1 volt. Magnetic flux is signed: choose a normal direction by right-hand rule with the loop's positive circulation, then flux is positive when B has a component along that normal. Faraday's law cares only about how this number changes in time, not its instantaneous value.

Why the minus sign (Lenz's law)?

The minus sign in EMF = −dΦ_B/dt is Lenz's law: the induced EMF drives a current whose own magnetic flux opposes the change in external flux. It is required by energy conservation. If the induced current's flux reinforced the external change, the system would amplify itself indefinitely from no input — a perpetual motion machine. Instead, you must do mechanical work against the back-EMF to push a magnet into a coil, and that mechanical work is exactly what becomes electrical energy in the circuit. Lenz isn't a separate law — it's the sign convention that makes Faraday's law obey the first law of thermodynamics.

What is motional EMF (moving rod in B)?

Motional EMF arises when a conductor moves through a static magnetic field — the charge carriers inside experience a Lorentz force qv × B that drives them along the conductor. For a rod of length L moving with velocity v perpendicular to a uniform B, the EMF between its ends is ε = BLv. If the rod slides on rails forming a closed circuit, this EMF drives a current. From the loop perspective, the area enclosed is changing at rate Lv, so dΦ_B/dt = BLv, and Faraday's law gives the same answer ε = BLv. The two viewpoints — Lorentz force on charges versus changing flux — are different but consistent ways to compute the same physics.

How does a transformer use Faraday's law?

A transformer is two coils — primary (N₁ turns) and secondary (N₂ turns) — wrapped around a shared iron core that channels magnetic flux between them. AC voltage on the primary creates oscillating flux Φ(t) in the core. Faraday's law applies separately to each coil: EMF₁ = −N₁ dΦ/dt and EMF₂ = −N₂ dΦ/dt. The shared dΦ/dt cancels, leaving V₂/V₁ = N₂/N₁ — the turn ratio. Power conservation (assuming an ideal transformer) gives I₂/I₁ = N₁/N₂. Step-up the voltage (more secondary turns) and current drops proportionally; step-down (fewer secondary turns) and current rises. This is how grid voltage is raised to 230kV–500kV for transmission and stepped back down for distribution.

Why is wireless charging Faraday-based?

A Qi wireless charger or wireless EV pad puts a transmitter coil under the surface and a receiver coil inside the device. Driving AC at 100–200 kHz through the transmitter produces oscillating B that links the receiver coil. Faraday's law induces an EMF in the receiver, rectified to DC for charging. Coupling efficiency depends on coil alignment, separation distance (drops sharply beyond a few millimeters for Qi), and operating frequency. Modern Qi 2.0 hits 75% end-to-end efficiency at 5–15 W. Resonant inductive coupling — both coils tuned to the same frequency — extends practical range and is the basis of mid-range wireless power and emerging EV pads up to 11 kW.

What is back-EMF in motors?

When a motor's armature spins in its magnetic field, Faraday's law makes it act simultaneously as a generator: rotation changes the flux through each winding and induces an EMF that opposes the applied voltage. This induced voltage is called back-EMF, and it scales linearly with rotation speed. Net current through the windings is (V_applied − V_back) / R, so as the motor accelerates, back-EMF grows and current drops. At no-load, back-EMF nearly equals applied voltage and current is small; at stall, back-EMF is zero and current spikes (which is why a stalled motor can burn out). Back-EMF lets you measure rotor speed without a tachometer and is the foundation of sensorless brushless DC motor control.